| L(s) = 1 | + 32·4-s + 113·9-s − 242·11-s + 768·16-s + 1.10e3·31-s + 3.61e3·36-s − 7.74e3·44-s + 4.80e3·49-s − 8.97e3·59-s + 1.63e4·64-s + 1.52e4·71-s + 6.20e3·81-s − 1.28e4·89-s − 2.73e4·99-s + 4.39e4·121-s + 3.53e4·124-s + 127-s + 131-s + 137-s + 139-s + 8.67e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.71e4·169-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.39·9-s − 2·11-s + 3·16-s + 1.15·31-s + 2.79·36-s − 4·44-s + 2·49-s − 2.57·59-s + 4·64-s + 3.01·71-s + 0.946·81-s − 1.62·89-s − 2.79·99-s + 3·121-s + 2.30·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.18·144-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.386773552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.386773552\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 113 T^{2} + p^{8} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 531793 T^{2} + p^{8} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 553 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 716447 T^{2} + p^{8} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6080638 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15265438 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4487 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 19806767 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7607 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6433 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 81155713 T^{2} + p^{8} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.25385442098684808435493857850, −11.03771497418561583949254898016, −10.51619324122315104439536883842, −10.21396999967025797202198621344, −9.927258676642024718476898201559, −9.217694492191265006926134726396, −8.238648359815476119327792480026, −7.953503403582154368538145605326, −7.52925756103424271863462170700, −7.07976095806000918181008875622, −6.66774752486851353787047677264, −6.05572951271251797397153161030, −5.52365790096618023658209588277, −4.94371530118119698849205204261, −4.18719453097482362243478779751, −3.32862403462340067701139247142, −2.70043049797326451669861539220, −2.25071148363607279130094471492, −1.53989123567351241179741824691, −0.71885572169458999409504553346,
0.71885572169458999409504553346, 1.53989123567351241179741824691, 2.25071148363607279130094471492, 2.70043049797326451669861539220, 3.32862403462340067701139247142, 4.18719453097482362243478779751, 4.94371530118119698849205204261, 5.52365790096618023658209588277, 6.05572951271251797397153161030, 6.66774752486851353787047677264, 7.07976095806000918181008875622, 7.52925756103424271863462170700, 7.953503403582154368538145605326, 8.238648359815476119327792480026, 9.217694492191265006926134726396, 9.927258676642024718476898201559, 10.21396999967025797202198621344, 10.51619324122315104439536883842, 11.03771497418561583949254898016, 11.25385442098684808435493857850
